Contact-Implicit Planning and Control for Non-Prehensile Manipulation Using State-Triggered Constraints

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Contact-Implicit Planning and Control for

Non-Prehensile Manipulation Using

State-Triggered Constraints

Maozhen Wang∗1†, Aykut ¨

Ozg¨un ¨

Onol∗1‡Philip Long2, and Ta¸skın Padır1∗

1Institute for Experiential Robotics, Northeastern University, Boston, MA, USA

2Atlantic Technological University, Galway, Ireland

Abstract. We present a contact-implicit planning approach that can

generate contact-interaction trajectories for non-prehensile manipulation

problems without tuning or a tailored initial guess and with high success

rates. This is achieved by leveraging the concept of state-triggered con-

straints (STCs) to capture the hybrid dynamics induced by discrete con-

tact modes without explicitly reasoning about the combinatorics. STCs

enable triggering arbitrary constraints by a strict inequality condition in

a continuous way. We ﬁrst use STCs to develop an automatic contact

constraint activation method to minimize the eﬀective constraint space

based on the utility of contact candidates for a given task. Then, we in-

troduce a re-formulation of the Coulomb friction model based on STCs

that is more eﬃcient for the discovery of tangential forces than the well-

studied complementarity constraints-based approach. Last, we include

the proposed friction model in the planning and control of quasi-static

planar pushing. The performance of the STC-based contact activation

and friction methods is evaluated by extensive simulation experiments

in a dynamic pushing scenario. The results demonstrate that our meth-

ods outperform the baselines based on complementarity constraints with

a signiﬁcant decrease in the planning time and a higher success rate.

We then compare the proposed quasi-static pushing controller against a

mixed-integer programming-based approach in simulation and ﬁnd that

our method is computationally more eﬃcient and provides a better track-

ing accuracy, with the added beneﬁt of not requiring an initial control

trajectory. Finally, we present hardware experiments demonstrating the

usability of our framework in executing complex trajectories in real-time

even with a low-accuracy tracking system.

Keywords: contact modeling, manipulation planning, optimization

∗Indicates equal contribution

†Currently at Amazon Robotics, MA, USA

‡Currently at Toyota Research Institute, Cambridge, MA, USA

∗This material is based upon work supported by the National Science Foundation

under Award No. 1928654. The authors would like to thank Dr. Michael Szmuk, Dr.

Taylor P. Reynolds, and Prof. Beh¸cet A¸cıkme¸se of the Autonomous Controls Lab at

the University of Washington for their useful feedback and insights.

arXiv:2210.09540v1 [cs.RO] 18 Oct 2022

2 M. Wang, A. ¨

Onol, P. Long, & T. Padır

1 Introduction

Non-prehensile manipulation will be a key capability for robots in both home

and industrial environments [40]. However, there is still a lack of reliable methods

that can compose contact-rich motions given only a high-level goal. The main

challenge is that the planning and control of contact interactions require discrete

decisions concerning the time and location of contact mode transitions that im-

pose switching constraints altering the evolution of the system dynamics. To

avoid predeﬁned contact schedules, a hybrid problem using mixed-integer pro-

gramming (MIP) may be solved, however, explicit modeling can be prohibitive

as the complexity grows exponentially with discrete variables [33]. Fortunately,

many discrete elements can be expressed in terms of continuous variables – such

as the distance, velocity, and force in the contact frame – by using complemen-

tarity constraints (CCs) which provide an eﬃcient way of including bi-directional

conditions in programs without the combinatorial complexity of explicitly doing

so. Thus, contact dynamics is deﬁned as a smooth optimization problem with

CCs in many physics engines and planning algorithms [38,49,33]. Yet, describing

the rich discrete aspects of contact-related problems using only bi-directional

statements is restrictive and may be ineﬃcient. We hypothesize that a continu-

ous model that implies uni-directional if conditions would be useful as it would

provide a more modular building block. Hence, we propose the use of state-

triggered constraints (STCs), ﬁrst put forward by Szmuk et al. [43], and analyze

their performance for contact-implicit planning (CIP) through non-prehensile

manipulation examples.

1.1 Contributions

The main contributions of our work are:

–An automatic contact constraint activation (CA) method that triggers only

the contact constraints useful for given task, thus reducing the problem’s

sensitivity to the number of contact pairs and enabling automatic contact

candidate assignment, i.e., all perceived surfaces can be assigned as contact

candidates as they are handled eﬃciently in the optimization. Extensive

simulations show that the proposed CA method improves the computational

eﬃciency and the success rate.

–We reformulate the Coulomb friction model using STCs and compare it to

a baseline based on CCs for discovering tangential forces to rotate a box

to random goal orientations. The proposed model is signiﬁcantly faster and

more robust than the baseline.

–The STC-based friction model is used to model the contact modes of planar

quasi-static pushing and compared to a MIP-based controller. Our method

outperforms the baseline both in speed and accuracy. Moreover, the method

can plan contact-interaction trajectories given only a desired state trajec-

tory without any heuristics. Although our method may run slower than the

Contact-Implicit Planning and Control of Pushing Using STCs 3

learning-based variant [15], we demonstrate experimentally that our STC-

based control can plan and track complex trajectories at a frequency high

enough for real-time applications.

To the best of our knowledge, this is the ﬁrst usage of STCs in contact modeling.

We show that STCs hold immense promise for modeling discrete elements in

this domain and can mitigate the need for explicit combinatorics and heuristics.

Our proposed CA and friction methods serve as example applications for STCs.

The extensive simulation and hardware experiments prove the eﬃcacy of the

proposed methodology.

1.2 Related Work

Mixed-Integer Programming The planning and control of contact-rich mo-

tions can be achieved by building a hybrid problem with explicit discrete vari-

ables and then solved by a MIP approach. In [7,1], MIP-based planners are devel-

oped and applied to locomotion on uneven terrains. In manipulation, MIP [14,2]

and exhausted tree search [8] have been used to achieve desired object behaviors.

Nevertheless, these approaches typically require simpliﬁcations and/or heuristics

to make them computationally tractable. [15,22,9] introduced various initializa-

tion strategies to reduce the computational burden, mostly with data-driven

techniques.

Contact-Implicit Trajectory Optimization Alternatively, contact-interaction

trajectories can be planned by incorporating a diﬀerentiable contact model into

a trajectory optimization framework, i.e., contact-implicit trajectory optimiza-

tion (CITO). [26,25] have shown complex contact-rich behaviors can be synthe-

sized for animated characters using convex optimization with soft constraints

by sacriﬁcing physical realism. In robotics applications, where motions must be

physically accurate, CCs are widely used to model inelastic rigid-body contacts

with Coulomb friction. In [49] the non-smooth trajectory optimization problem

with impacts and discontinuities are transcribed into a bi-level program with

CCs. Posa et al. [33] solved this problem simultaneously using direct collocation.

Similar methods have been proposed for planar manipulation [10] and dynamic

pushing [37]. In [24,21], hierarchical strategies using warm-starting to improve

the computational eﬃciency are presented, while [19,32] propose methods to

improve the integration accuracy of such numerical schemes. Other works have

focused on replacing CCs for more eﬃcient direct programs, e.g., [48,6,39].

The optimization problem can be solved faster than direct optimization via

Hessian-approximating diﬀerential dynamic programming (DDP) variants, such

as iterative linear quadratic regulator (iLQR) [17], typically with smooth contact

models as these approaches cannot easily handle constraints. In [44,27], smoother

fragments of CCs are used with DDP variants, and the potential to run CITO as

model predictive control (MPC) is demonstrated. [28] utilized an explicit smooth

contact model to enable real-time MPC for highly-dynamic quadruped motions.

4 M. Wang, A. ¨

Onol, P. Long, & T. Padır

However, these methods usually require either a good initial guess and/or te-

dious tuning due to the unreliable convergence of DDP methods. Recently, we

proposed a CITO framework based on a variable smooth contact model [30]

that decouples the relaxation from the contact model by injecting smooth vir-

tual forces into the underactuated dynamics with frictional contacts. Hence, both

nonlinear programming and DDP variants can be used to solve the problem, as

shown by [20,31,46,29]. While the methods developed here are applicable to our

previous work, we build upon a CC-based CITO framework similar to [33] as it is

more widely used in the related literature and thus a more relevant comparison.

Planar Pushing The mechanics of planar pushing in the presence of friction

was ﬁrst studied in [23] which focuses on the rotational direction and the ob-

ject’s center. In [18], the mechanics for stable pushing is studied. More recently,

[50] have shown that with a sticking contact, Dubins path can plan trajectories

eﬃciently. In [14] a hybrid MPC framework is proposed to track a pushing trajec-

tory without limiting the contact mode to sticking only, instead the contact mode

for each time step is deﬁned as a discrete variable. To avoid the combinatorial

problem’s computational, convex problems are solved for a fraction of potential

contact schedules for the prediction horizon, and the solution is determined com-

paratively. In [15], an aggregated contact mode selection method is proposed to

reduce the complexity of MIP and to eliminate the need for a predeﬁned contact

mode schedule. They proposed a learning method to predict the proper contact

mode to achieve a higher control frequency. However, this approach is not very

versatile as it requires re-training for object shape variations.

State-Triggered Constraints STCs were ﬁrst introduced in [43] as a more

modular, uni-directional alternative to CCs and were used to avoid MIP and

enable real-time performance for a rocket landing task with state-dependent

constraints. In [42], this concept is extended into a more general form called com-

pound STCs showing that more articulate trigger and constraint conditions can

be obtained by applying Boolean logic operations. STCs have been successfully

applied to real-time rocket landing [34,35] and quadrotor path planning [41].

2 Background

2.1 Contact Model

Following [38], a contact impulse due to a frictionless inelastic collision between

two rigid bodies can be modeled by 0 ≤φ(q)⊥λn≥0; where the non-negativity

condition for the signed distance φ(q) prevents interpenetration between bodies,

and the non-negativity condition for the normal contact force λnensures that

the contact bodies can only push each other. The frictional forces λt∈R2

can be modeled using the relative tangential (slip) velocity in the contact frame

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Contact-ImplicitPlanningandControlforNon-PrehensileManipulationUsingState-TriggeredConstraintsMaozhenWang*1,AykutOzgunOnol1PhilipLong2,andTasknPadr1*1InstituteforExperientialRobotics,NortheasternUniversity,Boston,MA,USA2AtlanticTechnologicalUniversity,Galway,IrelandAbstract.Wepresentacontac...

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Contact-Implicit Planning and Control for Non-Prehensile Manipulation Using State-Triggered Constraints

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